In analytic geometry, a coordinate system is defined as as pair $(O, \mathcal{B})$ consisting of a point $O$ in the space, called the origin, and a basis $\mathcal{B}=(\overrightarrow{b}_1, \overrightarrow{b}_2, \overrightarrow{b}_3)$ for the space of free vectors.
Since $\mathcal{B}$ is a basis for the space of free vector, every free vector $\overrightarrow{v}$ might be written uniquely as:
$$\overrightarrow{v}=v_1\cdot \overrightarrow{b}_1+v_2\cdot \overrightarrow{b}_2+v_3\cdot \overrightarrow{b}_3.$$ In particular, we have coordinates $v_1, v_2$ and $v_3$ for $\overrightarrow{v}$ with respect $\mathcal{B}$, which I write as $$\overrightarrow{v}=(v_1, v_2, v_3)_{\mathcal{B}}.$$
What is really the role of the origin in the coordinate system?
I mean, once we choose a basis we already have a bijection of the space of free vectors to $\mathbb R^3$.
I know that once we fix the origin $O$ we can find the coordinate of any vector $\overrightarrow{AB}$ in terms of the coordinates of $\overrightarrow{OA}$ and $\overrightarrow{OB}$, for:
$$\overrightarrow{OA}=(a_1, a_2, a_3)_{\mathcal{B}}\ \textrm{and}\ \overrightarrow{OB}=(b_1, b_2, b_3)_{\mathcal{B}}\implies \overrightarrow{OA}=\overrightarrow{OB}-\overrightarrow{OA}=(b_1-a_1, b_2-a_2, b_3-a_3)_{\mathcal{B}}.$$ I believe there is a conceptual gap in my understading of the real meaning of a coordinate system. So, what is the theoretical importance of fixing an origin after all?
Maybe what I'm missing is that the idea of the coordinate system is to establish a bijection between $\mathbb E^3$ (the space of euclidean geometry) and $\mathbb R^3$. In order to do that, we could define:
$$\varphi_{O, \mathcal{B}}: \mathbb E^3\rightarrow \mathbb R^3$$ setting
$$\varphi_{O, \mathcal{B}}(P)=(p_1, p_2, p_3)$$ where $\overrightarrow{OP}=(p_1, p_2, p_3)_{\mathcal{B}}$. Is that it? And, in fact, for defining this bijection is necessary to fix the point $O$.
You've hit upon the distinction between vector spaces and affine spaces. If you identify vectors with points, the zero vector is identified with the origin. But we could translate all these identifications to change the origin, thereby preserving the paths between points. An affine space identifies each vector with the set of such paths that involve the same displacement; for example, the zero vector is identified with the set of point-to-same-point, zero-length paths.
You can think of these sets as equivalence classes labelled with vectors. With respect to the equivalence relation thereby induced, our different choices of vector-to-point identifications are all equivalent; the only element of the resulting quotient space is the aforementioned affine space.